\documentclass[12pt]{article}

\usepackage[italian]{babel}
\usepackage{amsmath}

\title{Gruppo \textbf{lab01}}

\author{Elisa Barnini \and Luca Geloso \and Matteo Barbieri}

\begin{document}

\maketitle

Calcolo del gradiente:

\[
\nabla f(x) = 
\begin{pmatrix}
	\frac{\partial{f}}{\partial{x_1}} \\[0.8em]
	\frac{\partial{f}}{\partial{x_2}}
\end{pmatrix}
\]

dove

\begin{center}
\begin{tabular}{ l c l r}
  $\frac{\partial{f}}{\partial{x_1}} $ & = & $2h(x_1^2 + x_2 - 11)(2x_1) + 2k(x_1 + x_2^2 - 7)$ & $ = $ \\
  & =  & $4hx_1(x_1^2 + x_2 - 11) + 2k(x_1 + x_2^2 - 7)$ &
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{ l c l r}
  $\frac{\partial{f}}{\partial{x_2}} $ & = &  $2h(x_1^2 + x_2 - 11) + 2k(x_1 + x_2^2 - 7)(2x_2)$ & $ = $ \\
  & =  & $2h(x_1^2 + x_2 - 11) + 4kx_2(x_1 + x_2^2 - 7)$ &
\end{tabular}
\end{center}

\vspace*{1em}

Calcolo della matrice hessiana $f''(x)$:

\[
f''(x) = 
\begin{pmatrix}
	\frac{\partial^2{f}}{\partial{x_1}^2} & \frac{\partial^2{f}}{\partial{x_1}\partial{x_2}} \\[0.8em]
	\frac{\partial^2{f}}{\partial{x_2}\partial{x_1}} & \frac{\partial^2{f}}{\partial{x_2}^2}
\end{pmatrix}
\]

dove \\

\begin{center}
\begin{tabular}{ l c l r}
  $\frac{\partial^2{f}}{\partial{x_1}^2} $ & = &  $4hx_1(2x_1) + 4h(x_1^2 + x_2 - 11) + 2k$ & $ = $ \\
  & = & $8hx_1^2 + 4h(x_1^2 + x_2 - 11) + 2k$ & \\[0.8em]
  $\frac{\partial^2{f}}{\partial{x_1}\partial{x_2}} $ & = &  $4hx_1 + 4hx_2$ \\[0.8em]
  $\frac{\partial^2{f}}{\partial{x_2}\partial{x_1}}$ & $=$ & $\frac{\partial^2{f}}{\partial{x_1}\partial{x_2}}$ \\[0.8em]
  $\frac{\partial^2{f}}{\partial{x_2}^2} $ & = &  $2h + 4kx_2(2x_2) + 4k(x_1 + x_2^2 - 7)$ & $ = $ \\
  & = & $2h + 8kx_2^2 + 4k(x_1 + x_2^2 - 7)$ &
\end{tabular}
\end{center}

\end{document}